Lattices and reduced cells as points in 6-space and selection of Bravais lattice type by projections

Andrews, Larry C.; Bernstein, H. J.

doi:10.1107/s0108767388006427

Cited by 48 publications

(73 citation statements)

References 14 publications

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“…The 44 lattice types are then characterized by additional equality relations among the six components of the reduced-cell metric tensor. Andrews & Bernstein (1988) describe unit cells as vectors in a Euclidean six-dimensional space and define each lattice type by its characteristic linear subspace. This formulation has been used to measure the similarity of a given six-dimensional cell vector with each lattice type: it is the length of the component perpendicular to the subspace corresponding to each lattice type (Paciorek & Bonin, 1992;Andrews & Bernstein, 1988).…”

Section: Bravais-lattice Determinationmentioning

confidence: 99%

“…Andrews & Bernstein (1988) describe unit cells as vectors in a Euclidean six-dimensional space and define each lattice type by its characteristic linear subspace. This formulation has been used to measure the similarity of a given six-dimensional cell vector with each lattice type: it is the length of the component perpendicular to the subspace corresponding to each lattice type (Paciorek & Bonin, 1992;Andrews & Bernstein, 1988). Any primitive triclinic cell describing a given lattice can be converted into such a reduced cell (Kfiv~, & Gruber, 1976;Andrews & Bernstein, 1988).…”

Section: Bravais-lattice Determinationmentioning

confidence: 99%

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Automatic processing of rotation diffraction data from crystals of initially unknown symmetry and cell constants

Kabsch¹

1993

J Appl Crystallogr

3,461

2,824

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An algorithm has been developed for the automatic interpretation of a given set of observed reciprocal-lattice points. It extracts a reduced cell and assigns indices to each reflection by a graph-theoretical implementation of the local indexing method. All possible symmetries of the observed lattice compatible with the metric of the reduced cell are recognized and reported, together with the unit-cell constants and the linear index transformation relating the conventional to the reduced cell. This algorithm has been incorporated into the program XDS [Kabsch (1988). J. Appl. Cryst. 21, 916-924], which is now able to process single-crystal area-detector data without prior knowledge of the symmetry and the unit-cell constants

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Section: Bravais-lattice Determinationmentioning

confidence: 99%

Section: Bravais-lattice Determinationmentioning

confidence: 99%

Automatic processing of rotation diffraction data from crystals of initially unknown symmetry and cell constants

Kabsch¹

1993

J Appl Crystallogr

3,461

2,824

View full text Add to dashboard Cite

show abstract

“…Such a reduced cell (in the Delaunay/ Niggli sense) can then be used for the proper crystallographic analysis of the crystal, beginning with the correct assignment of its Bravais lattice, and providing a normalized description, with a view to comparison. At this last step, the kind of supercell reduction that we have developed thus connects with established Delaunay/Niggli-like reduction theory, and we direct the reader to the literature on this subject and its relation to the reduction of ternary quadratic forms for further insight (see, for example, Burzlaff et al, 2002;Andrews & Bernstein, 1988;Gruber, 1989;Oishi-Tomiyasu, 2012;Andrews & Bernstein, 2014, and references therein).…”

Section: Some Comments On the Methods Of Reductionmentioning

confidence: 87%

Lattice reduction using a Euclidean algorithm

Mújica

2017

Acta Cryst Sect A

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The need to reduce a periodic structure given in terms of a large supercell and associated lattice generators arises frequently in different fields of application of crystallography, in particular in the ab initio theoretical modelling of materials at the atomic scale. This paper considers the reduction of crystals and addresses the reduction associated with the existence of a commensurate translation that leaves the crystal invariant, providing a practical scheme for it. The reduction procedure hinges on a convenient integer factorization of the full period of the cycle (or grid) generated by the repeated applications of the invariant translation, and its iterative reduction into sub-cycles, each of which corresponds to a factor in the decomposition of the period. This is done in successive steps, each time solving a Diophantine linear equation by means of a Euclidean reduction algorithm in order to provide the generators of the reduced lattice.

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“…The diffraction images (Fig. 3) were processed by a semiautomated protocol using the DPS algorithm (Steller et al, 1997) and least-squares ®tting (Andrews & Bernstein, 1988;Paciorek & Bonin, 1992). The crystal belonged to the triclinic P1 space group, with the unit-cell parameters and data-collection statistics summarized in Table 1.…”

Section: Resultsmentioning

confidence: 99%

Crystallization and preliminary X-ray crystallographic analysis of 2-methyl-3-hydroxypyridine-5-carboxylic acid (MHPC) oxygenase fromPseudomonassp. MA-1

et al. 2005

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2-Methyl-3-hydroxypyridine-5-carboxylic acid (MHPC) oxygenase (MHPCO) catalyzes the conversion of an aromatic substrate, MHPC, to an aliphatic compound, -(N-acetylaminomethylene)-succinic acid, and is involved in the degradation of vitamin B 6 by the soil bacterium Pseudomonas sp. MA-1. Using only FAD as a cofactor, MHPCO is unique in catalyzing hydroxylation and subsequent aromatic ring cleavage without requiring a metal-ion cofactor. Here, the crystallization of MHPCO is reported together with preliminary X-ray crystallographic data. An MHPCO crystal obtained by hanging-drop vapour diffusion diffracted X-rays to 2.25 A Ê resolution and belonged to the triclinic space group P1, with four molecules per asymmetric unit.

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Lattices and reduced cells as points in 6-space and selection of Bravais lattice type by projections

Cited by 48 publications

References 14 publications

Automatic processing of rotation diffraction data from crystals of initially unknown symmetry and cell constants

Automatic processing of rotation diffraction data from crystals of initially unknown symmetry and cell constants

Lattice reduction using a Euclidean algorithm

Crystallization and preliminary X-ray crystallographic analysis of 2-methyl-3-hydroxypyridine-5-carboxylic acid (MHPC) oxygenase fromPseudomonassp. MA-1

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